Playback stopping with internet connectivity could be caused by the way you select the playlist. In the Music app, you can choose Playlists directly but underneath that is Downloaded Music, which under that has Playlists again. I use the Downloaded Music->Playlists when using offline to guarantee that the player is not trying to use my data or WiFi for checking the cloud status of the song I am listening to. I think the failure to connect to the cloud is causing the playback to stop. See if using Downloaded Music helps.
Yes you can select multiple songs in iTunes and drag and drop into a playlist.
For a Mac or Windows, selecting a bunch of songs together (like an artist) click the first song, then hold Shift and click the last song and all of them will be selected. Drag and drop any of the selected songs to the playlist.
For Mac and songs that are not in order or only a few are wanted, click the first song, then hold Command (Windows use Ctrl) then click the next songs you want. Drag and drop any of the selected songs to the playlist.